Step | Hyp | Ref
| Expression |
1 | | erdsze2lem.n |
. . . . 5
⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1)) |
2 | | erdsze2.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ℕ) |
3 | | nnm1nn0 12272 |
. . . . . . 7
⊢ (𝑅 ∈ ℕ → (𝑅 − 1) ∈
ℕ0) |
4 | 2, 3 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑅 − 1) ∈
ℕ0) |
5 | | erdsze2.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ℕ) |
6 | | nnm1nn0 12272 |
. . . . . . 7
⊢ (𝑆 ∈ ℕ → (𝑆 − 1) ∈
ℕ0) |
7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑆 − 1) ∈
ℕ0) |
8 | 4, 7 | nn0mulcld 12296 |
. . . . 5
⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) ∈
ℕ0) |
9 | 1, 8 | eqeltrid 2845 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
10 | | nn0p1nn 12270 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ) |
11 | 9, 10 | syl 17 |
. . 3
⊢ (𝜑 → (𝑁 + 1) ∈ ℕ) |
12 | | erdsze2.f |
. . . 4
⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) |
13 | | erdsze2lem.g |
. . . 4
⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))–1-1→𝐴) |
14 | | f1co 6679 |
. . . 4
⊢ ((𝐹:𝐴–1-1→ℝ ∧ 𝐺:(1...(𝑁 + 1))–1-1→𝐴) → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))–1-1→ℝ) |
15 | 12, 13, 14 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝐹 ∘ 𝐺):(1...(𝑁 + 1))–1-1→ℝ) |
16 | 9 | nn0red 12292 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℝ) |
17 | 16 | ltp1d 11903 |
. . . 4
⊢ (𝜑 → 𝑁 < (𝑁 + 1)) |
18 | 1, 17 | eqbrtrrid 5115 |
. . 3
⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (𝑁 + 1)) |
19 | 11, 15, 2, 5, 18 | erdsze 33158 |
. 2
⊢ (𝜑 → ∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))))) |
20 | | velpw 4544 |
. . . 4
⊢ (𝑡 ∈ 𝒫 (1...(𝑁 + 1)) ↔ 𝑡 ⊆ (1...(𝑁 + 1))) |
21 | | imassrn 5978 |
. . . . . . . 8
⊢ (𝐺 “ 𝑡) ⊆ ran 𝐺 |
22 | | f1f 6667 |
. . . . . . . . . 10
⊢ (𝐺:(1...(𝑁 + 1))–1-1→𝐴 → 𝐺:(1...(𝑁 + 1))⟶𝐴) |
23 | 13, 22 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))⟶𝐴) |
24 | 23 | frnd 6605 |
. . . . . . . 8
⊢ (𝜑 → ran 𝐺 ⊆ 𝐴) |
25 | 21, 24 | sstrid 3937 |
. . . . . . 7
⊢ (𝜑 → (𝐺 “ 𝑡) ⊆ 𝐴) |
26 | | erdsze2.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
27 | | reex 10961 |
. . . . . . . . 9
⊢ ℝ
∈ V |
28 | | ssexg 5251 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧ ℝ
∈ V) → 𝐴 ∈
V) |
29 | 26, 27, 28 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ V) |
30 | | elpw2g 5272 |
. . . . . . . 8
⊢ (𝐴 ∈ V → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴)) |
31 | 29, 30 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ↔ (𝐺 “ 𝑡) ⊆ 𝐴)) |
32 | 25, 31 | mpbird 256 |
. . . . . 6
⊢ (𝜑 → (𝐺 “ 𝑡) ∈ 𝒫 𝐴) |
33 | 32 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ 𝒫 𝐴) |
34 | | vex 3435 |
. . . . . . . . . . . 12
⊢ 𝑡 ∈ V |
35 | 34 | f1imaen 8783 |
. . . . . . . . . . 11
⊢ ((𝐺:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡) |
36 | 13, 35 | sylan 580 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ≈ 𝑡) |
37 | | fzfid 13689 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ∈ Fin) |
38 | | simpr 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ⊆ (1...(𝑁 + 1))) |
39 | | ssfi 8936 |
. . . . . . . . . . . . 13
⊢
(((1...(𝑁 + 1))
∈ Fin ∧ 𝑡 ⊆
(1...(𝑁 + 1))) → 𝑡 ∈ Fin) |
40 | 37, 38, 39 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝑡 ∈ Fin) |
41 | | enfii 8952 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ Fin ∧ (𝐺 “ 𝑡) ≈ 𝑡) → (𝐺 “ 𝑡) ∈ Fin) |
42 | 40, 36, 41 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 “ 𝑡) ∈ Fin) |
43 | | hashen 14057 |
. . . . . . . . . . 11
⊢ (((𝐺 “ 𝑡) ∈ Fin ∧ 𝑡 ∈ Fin) → ((♯‘(𝐺 “ 𝑡)) = (♯‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡)) |
44 | 42, 40, 43 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((♯‘(𝐺 “ 𝑡)) = (♯‘𝑡) ↔ (𝐺 “ 𝑡) ≈ 𝑡)) |
45 | 36, 44 | mpbird 256 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (♯‘(𝐺 “ 𝑡)) = (♯‘𝑡)) |
46 | 45 | breq2d 5091 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ↔ 𝑅 ≤ (♯‘𝑡))) |
47 | 46 | biimprd 247 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑅 ≤ (♯‘𝑡) → 𝑅 ≤ (♯‘(𝐺 “ 𝑡)))) |
48 | | erdsze2lem.i |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺)) |
49 | 48 | ad2antrr 723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺)) |
50 | 38 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (1...(𝑁 + 1))) |
51 | | simprl 768 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑡) |
52 | 50, 51 | sseldd 3927 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (1...(𝑁 + 1))) |
53 | | simprr 770 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡) |
54 | 50, 53 | sseldd 3927 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (1...(𝑁 + 1))) |
55 | | isorel 7191 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺) ∧ (𝑥 ∈ (1...(𝑁 + 1)) ∧ 𝑦 ∈ (1...(𝑁 + 1)))) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦))) |
56 | 49, 52, 54, 55 | syl12anc 834 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 ↔ (𝐺‘𝑥) < (𝐺‘𝑦))) |
57 | 56 | biimpd 228 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) ∧ (𝑥 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
58 | 57 | ralrimivva 3117 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
59 | | elfznn 13282 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℕ) |
60 | 59 | nnred 11986 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ (1...(𝑁 + 1)) → 𝑡 ∈ ℝ) |
61 | 60 | ssriv 3930 |
. . . . . . . . . . . . . 14
⊢
(1...(𝑁 + 1))
⊆ ℝ |
62 | 61 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (1...(𝑁 + 1)) ⊆ ℝ) |
63 | | ltso 11054 |
. . . . . . . . . . . . 13
⊢ < Or
ℝ |
64 | | soss 5523 |
. . . . . . . . . . . . 13
⊢
((1...(𝑁 + 1))
⊆ ℝ → ( < Or ℝ → < Or (1...(𝑁 + 1)))) |
65 | 62, 63, 64 | mpisyl 21 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or (1...(𝑁 + 1))) |
66 | 26 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐴 ⊆ ℝ) |
67 | | soss 5523 |
. . . . . . . . . . . . 13
⊢ (𝐴 ⊆ ℝ → ( <
Or ℝ → < Or 𝐴)) |
68 | 66, 63, 67 | mpisyl 21 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → < Or 𝐴) |
69 | 23 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → 𝐺:(1...(𝑁 + 1))⟶𝐴) |
70 | | soisores 7192 |
. . . . . . . . . . . 12
⊢ ((( <
Or (1...(𝑁 + 1)) ∧ <
Or 𝐴) ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1)))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
71 | 65, 68, 69, 38, 70 | syl22anc 836 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
72 | 58, 71 | mpbird 256 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡))) |
73 | | isocnv 7195 |
. . . . . . . . . 10
⊢ ((𝐺 ↾ 𝑡) Isom < , < (𝑡, (𝐺 “ 𝑡)) → ◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡)) |
74 | 72, 73 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡)) |
75 | | isotr 7201 |
. . . . . . . . . 10
⊢ ((◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))) |
76 | 75 | ex 413 |
. . . . . . . . 9
⊢ (◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
77 | 74, 76 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
78 | | resco 6152 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∘ 𝐺) ↾ 𝑡) = (𝐹 ∘ (𝐺 ↾ 𝑡)) |
79 | 78 | coeq1i 5766 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ◡(𝐺 ↾ 𝑡)) |
80 | | coass 6167 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∘ (𝐺 ↾ 𝑡)) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) |
81 | 79, 80 | eqtri 2768 |
. . . . . . . . . . 11
⊢ (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) |
82 | | f1ores 6727 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡)) |
83 | 13, 82 | sylan 580 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡)) |
84 | | f1ococnv2 6739 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ↾ 𝑡):𝑡–1-1-onto→(𝐺 “ 𝑡) → ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡))) |
85 | 83, 84 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = ( I ↾ (𝐺 “ 𝑡))) |
86 | 85 | coeq2d 5769 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) = (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡)))) |
87 | | coires1 6166 |
. . . . . . . . . . . 12
⊢ (𝐹 ∘ ( I ↾ (𝐺 “ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡)) |
88 | 86, 87 | eqtrdi 2796 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝐹 ∘ ((𝐺 ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡))) = (𝐹 ↾ (𝐺 “ 𝑡))) |
89 | 81, 88 | eqtrid 2792 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡))) |
90 | | isoeq1 7182 |
. . . . . . . . . 10
⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
91 | 89, 90 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
92 | | imaco 6153 |
. . . . . . . . . 10
⊢ ((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) |
93 | | isoeq5 7186 |
. . . . . . . . . 10
⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
94 | 92, 93 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) |
95 | 91, 94 | bitrdi 287 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
96 | 77, 95 | sylibd 238 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
97 | 47, 96 | anim12d 609 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
98 | 45 | breq2d 5091 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ↔ 𝑆 ≤ (♯‘𝑡))) |
99 | 98 | biimprd 247 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (𝑆 ≤ (♯‘𝑡) → 𝑆 ≤ (♯‘(𝐺 “ 𝑡)))) |
100 | | isotr 7201 |
. . . . . . . . . 10
⊢ ((◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡))) |
101 | 100 | ex 413 |
. . . . . . . . 9
⊢ (◡(𝐺 ↾ 𝑡) Isom < , < ((𝐺 “ 𝑡), 𝑡) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
102 | 74, 101 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
103 | | isoeq1 7182 |
. . . . . . . . . 10
⊢ ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
104 | 89, 103 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)))) |
105 | | isoeq5 7186 |
. . . . . . . . . 10
⊢ (((𝐹 ∘ 𝐺) “ 𝑡) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
106 | 92, 105 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) |
107 | 104, 106 | bitrdi 287 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((((𝐹 ∘ 𝐺) ↾ 𝑡) ∘ ◡(𝐺 ↾ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), ((𝐹 ∘ 𝐺) “ 𝑡)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
108 | 102, 107 | sylibd 238 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)) → (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
109 | 99, 108 | anim12d 609 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → ((𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) → (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
110 | 97, 109 | orim12d 962 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))) |
111 | | fveq2 6769 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → (♯‘𝑠) = (♯‘(𝐺 “ 𝑡))) |
112 | 111 | breq2d 5091 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑅 ≤ (♯‘𝑠) ↔ 𝑅 ≤ (♯‘(𝐺 “ 𝑡)))) |
113 | | reseq2 5884 |
. . . . . . . . . 10
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡))) |
114 | | isoeq1 7182 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)))) |
115 | 113, 114 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)))) |
116 | | isoeq4 7185 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)))) |
117 | | imaeq2 5963 |
. . . . . . . . . 10
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡))) |
118 | | isoeq5 7186 |
. . . . . . . . . 10
⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
119 | 117, 118 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
120 | 115, 116,
119 | 3bitrd 305 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
121 | 112, 120 | anbi12d 631 |
. . . . . . 7
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
122 | 111 | breq2d 5091 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → (𝑆 ≤ (♯‘𝑠) ↔ 𝑆 ≤ (♯‘(𝐺 “ 𝑡)))) |
123 | | isoeq1 7182 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ 𝑠) = (𝐹 ↾ (𝐺 “ 𝑡)) → ((𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))) |
124 | 113, 123 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))) |
125 | | isoeq4 7185 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)))) |
126 | | isoeq5 7186 |
. . . . . . . . . 10
⊢ ((𝐹 “ 𝑠) = (𝐹 “ (𝐺 “ 𝑡)) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
127 | 117, 126 | syl 17 |
. . . . . . . . 9
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
128 | 124, 125,
127 | 3bitrd 305 |
. . . . . . . 8
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)) ↔ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))) |
129 | 122, 128 | anbi12d 631 |
. . . . . . 7
⊢ (𝑠 = (𝐺 “ 𝑡) → ((𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))) ↔ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) |
130 | 121, 129 | orbi12d 916 |
. . . . . 6
⊢ (𝑠 = (𝐺 “ 𝑡) → (((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))) ↔ ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡))))))) |
131 | 130 | rspcev 3561 |
. . . . 5
⊢ (((𝐺 “ 𝑡) ∈ 𝒫 𝐴 ∧ ((𝑅 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))) ∨ (𝑆 ≤ (♯‘(𝐺 “ 𝑡)) ∧ (𝐹 ↾ (𝐺 “ 𝑡)) Isom < , ◡ < ((𝐺 “ 𝑡), (𝐹 “ (𝐺 “ 𝑡)))))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) |
132 | 33, 110, 131 | syl6an 681 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ⊆ (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))) |
133 | 20, 132 | sylan2b 594 |
. . 3
⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 (1...(𝑁 + 1))) → (((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))) |
134 | 133 | rexlimdva 3215 |
. 2
⊢ (𝜑 → (∃𝑡 ∈ 𝒫 (1...(𝑁 + 1))((𝑅 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡))) ∨ (𝑆 ≤ (♯‘𝑡) ∧ ((𝐹 ∘ 𝐺) ↾ 𝑡) Isom < , ◡ < (𝑡, ((𝐹 ∘ 𝐺) “ 𝑡)))) → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠)))))) |
135 | 19, 134 | mpd 15 |
1
⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) |